Chapter 13
U-Substitution and Integration
by Parts
I
n this chapter, you encounter some of the more advanced integration techniques:
u-substitution and integration by parts. You use u-substitution very, very often in integration problems. For many integration problems, consider starting with a u-substitution if you
don’t immediately know the antiderivative. Another common technique is integration by
parts, which comes from the product rule for derivatives. One of the difficult things about
these problems is that even when you know which procedure to use, you still have some
freedom in how to proceed; what to do isn’t always clear, so dive in and try different things.
The Problems You’ll Work On
This chapter is the start of more challenging integration problems. You work on the
following skills:
✓ Using u-substitution to find definite and indefinite integrals
✓ Using integration by parts to find definite and indefinite integrals
What to Watch Out For
Here are a few things to keep in mind while working on the problems in this chapter:
✓ Even if you know you should use a substitution, there may be different substitutions
to try. As a rule, start simple and make your substitution more complex if your first
choice doesn’t work.
✓ When using a u-substitution, don’t forget to calculate du, the differential.
✓ You can algebraically manipulate both du and the original u-substitution, so play
with both!
✓ For the integration by parts problems, if your pick of u and dv don’t seem to be
working, try switching them.